Contamination events
We simulate contamination events by adding a series of
anomalous water quality values to the observed background
data. These are added with a function that allows the sharpness
of the anomaly start and end along with the length and
amplitude of the anomaly to be controlled by the user (see
Murray et al. )
xE(t) ¼ xB(t) þ δ Eind σx Emax,
where xE(t) and xB(t) are the water quality signal in the event
and background forms, respectively. The parameters Eind
and Emax determine respectively the location and the deviation
from the background of the contamination. The
parameter δ is set to 1.0 or 1.0 to define the direction of
the signal deviation away from the background. Values of
Eind are taken from the Gaussian cumulative distribution
function (cdf) with a width specified in the number of time
steps to allow for a sharper or more gradual change from
and back to the background levels. The maximum deviation
from the background is given by σx Emax, where σx is the
standard deviation of the background signal. In the experiments
reported here, events are added to the background
at every 110 time steps (220 minutes) with a maximum
strength of 1, a total length of 16 time steps (32 minutes)
and a transition from background to full strength over six
time steps (12 minutes).
Tuning parameters
We present the tuning parameters that have been used, and
which allow the figures that will be presented in the next
subsection to be obtained. Recall that both SPRT and
CUSUM consider the knowledge of the pdf of the data.
Here, we use Gaussian pdf, where in the absence of contamination
the mean and variance of our data are
respectively given by μ0 and σ0, which are determined by
computing respectively the mean and standard deviation
of the signal of interest, and in the absence of contamination.
They change to μ1 and σ1 in the presence of
contamination, and in this study we consider that μ1 ¼ 5 μ0
and σ1 ¼ 30 σ0 for all the measurements except the TOC,
where μ1 ¼ 5 μ0 and σ1 ¼ 5 σ0. A and B were computed by
using Equation (5), and determine the thresholds for
SPRT, the threshold for the CUSUM test is given by h0.
We summarize the values of the tuning parameters for the
SPRT and CUSUM in Table 1. As mentioned previously,